Step 88 if NL(J) / IX(IM,1) then gojo step 86 fi Step 89 set L*— -L + 1; ID( J1, J) ITEST(lB(l),L)
Step 90 £3^° step 86
Step 100 (find how many factors have not been included in generators) set L*— 0: I#— 0
«V\A 7
Step 101 set I*— 1 + 1
Step 102 if I > LI then goto step 105 fi
Step 103 if IX(I, 1)^ 100 then set L *-L + IX(l,2) fi Step 104 goto step 101
Step 105 if L = 0 then goto step 200 fi■■■ .» ■ ii— ‘ a 11 ■ r*v_n Step 106 if L > 2 thep goto step 120 fi Step 107 set I*— 0
Step ■■ ■■ 108 set I*— 1 + 1
Step 109 if I ^ LI thep ggto step 200 fi
Step 110 if IX(l,4) / 0 then goto step 108 fi Step 111 set 114— IX(I,1); I2«-0j I3«-0
Step 112 set I2«*~I2 + 1
Step 113 i£ I2>IX(I,2) then goto Btep 108 fi
Step 114 sjit 13*— 13 + V
Step 115 if^ 13 )> N then^ goto step 108 fi
Step 116 if^ NL(I3) / 11 then goto step 114 j£i
Step 117 set NG*-NG + 1; ID(NG,I3)*— t; goto step 112 Step 120 (initialise for DEFGEN)
get NIG*- L; NVI+— 0
Step 121 set NVJ<— NVI + 1; MVl(NVl)<r- 0 Step 122 set MVl(NVl)*- I0NBT(MVl(NVl),NVl)
Step 123 A A . if NVI ^ NIG then goto step A A A * ■ i , ...121 fi
Step 125 (set up MASK to detect interactions)
set MASK*1 • 0; I*— 0 Step 126 set I*— I + 1
Step 127 if I > LI then goto step 135 fi
Step 128 if IX(l,4) £ 0 then goto step 126 fi Step 129 set I1*-IX(l,l)j 12*- 0
Step 131 Jf 12 > N then goto step 126 fi
Step 132 if NL(I2) / 11 then goto step 130 fi
Step 133 set MASK*— I0NBT(MASK, 12); goto step 130 fi Step 135 r8gt i*— N
Step 136 get I«— I + 1
Step 137 if I > NV then goto step 150 fi
Step 138 if and(MASK, MV(l)) = 0 then §££9 step 136 £i
Step 139 set 11*-0; J<— 0; NVI<— NVI + 1; MVl(NVl)*- 0 Step 140 get^ J*— J + 1
Step 1/4-1 if J > N tjien ggto step 136 fi
Step 142 if ITEST(MASK,J) = 1 t£en do steps 143. 144 od fi Step 143 s6t 11*— 11 + 1
Step 144 if ITEST(MV(l),J) = 1 t£en set MVl(NVI>6-I0NBT(MVl(NVl), 11) fi Step 145 step 140
Step 150 call DEFGEN (to produce generators from, an equivalent two level fractional factorial)
(enter with NIG, NVI, MVl(.)> return with NGI and IB(.))
Step 155 (set number of factor levels for entry to SELG) set I<— 0; I2<— 0
rv-''- ' Step 15^ set K — 1 + 1
Step 157 if I > N then goto step 170 fi Step 158 Sj5t_ I1<— 0
Step 139 set I1<— 11 + 1
Step 160 if 11 > LI then goto step 156 fi
S t e p 161 if I X ( I 1, 4 ) / 0 t h e n g o t o s t e p 1 5 9 fi
Step 162 if NL(I) / IX(I1,1) then goto step 159 fi
Step 163 jset I2<— 12 + 1; NLL(I2)*— NL(l); LF(I2)*— I; goto step 159 Step 170 call SELG (to select generators for multi-level asymmetric
factorial, given the equivalent generators for a two-level factorial)
(enter with NGI, IB(.)> NIG,NLL(.), LI (number of factor subsets), IX(. , .) (properties of factor subsets); return with IDD(. , .) (generators in integer form))
Step 180 Step 181 Step 182 Step 183 Step 184 Step 185 Step 186 Step 200 Step 201 Step 202 Step 203 Step 204 Step 205 Step 206 Step 207 Step 208 Step 209 Step 2090 Step 2091 Step 2092 Step 2093 Step 2094 Step 2093 Step 2096 Step 210 Step 211 Step 212 Step 213
(copy generators from SELG- into generators for all factors)
set I<— 0
set I<— I + 1; NG<— NG + 1
if I > NGI t-hen goto step 200 fi
set J*— 0 set — J + 1
if J > NIG tjien goto step 181 fi^
set 11 <— LF(J); ID(NG, I1)4— IDD(l, J); step 184 (find full design size NO = product of generator orders)
set 14— 0; N0«— t set I <— 1 + 1
if I > NG then goto step 208 fi set J<— 0
s££ J<- J + 1
if J > N then gojo^ step 207
set Il(j)<— J0RD(ID(I,J), NL(j)); goto step 204
se£ IL(l)<— LCM(lI, N); N04-N0*IL(l); goto step 201 set NFUL I*-1; 14— 0
se£ 14— I + 1; ,if I > N goto step 2091 fi
set NPULL4 - NFULL * NL(l); goto step 209 if N 0 ^ NFULL then goto step 210
elge g£t N04- NFULL; NG4— N; I*— 0 fi set 1 4 — I + 1i; if I > N tften goto step 210
else set J4-0; IL(l)4— NL(l) fi
set J4— J +1; if N then step 2092 fi set ID(I,J)4- 0
if I = J then set ID(l,J)*~1 fi goto steg 2093
print heading; set 14— 0
s_et 14 — I + 1; jLf I ^ NO t]ien gpto step 1000 fi
call LEV (to find levels of all factors for I'th observation) (enter with I, N, NG, IL, ID; return with IK)
grint N values of IK(.); goto step 211
10 / £ HIM) X x ^ ((lk(TL,V>?l)N- ^(n(rw)/ons' 100 NGI f t °- !<-£+•
w I - Ih So r«-r*t 1 3 W +T 1 >N<1 urg
I > LI
103
, fl,i) £lo< Uo •off I > L I no, in i*v IS > N in U o NK NVI 111 I IT I >LI tfo M#fK •<--- urC3 9 c
I > N V II* O J-*— o NVX«— N/r-M Mvr(N/r^«— o | H*— H -vl I MVI (NVI W— itMbt y frlVT(M»l), IQ IfO /~7 ^ u~ r€. 3 9 c( no S E L G J«— J-» t |
I<— It I
J > N
L E V looo
111
The Automatic Design of Experiments Some Practical Algorithms
CHAPTER SEVEN
REDUCING THE BALANCED ASYMMETRIC FRACTION
1 Background 2 Using the trace 3 D-optimal algorithms 4 Examples
1_____Background
The method developed in chapter six leads to the generation of balanced fractional designs which are adequate for estimating all specified main effects and interactions. Sometimes however
the generated designs are more than adequate: the number of observations exceeds the number of contrasts to be estimated by many more than the few extra needed for error estimation. In these cases the costs of practical experimentation dictate that the size of the design should be reduced further. In this
chapter I present two criteria that are widely applied in reducing experimental designs. See Fedorov (1972). The two criteria, are: A-optimality: The trace of the inverted information (cross-product) matrix is minimised. In the case of a discrete design, such as an asymmetric factorial using qualitative variables, the use of the A-optimality criterion means choosing a subset of r points from a design with n points (n> r) such that the trace of the inverted information matrix of the r-subdesign is no greater than that of any other r-subdesign.
D-optimality: The determinant of the inverted information matrix is minimised. In the case of a discrete design, the use of the D-optimality criterion means choosing a subset of r points from a design with n points (n > r) such that the determinant of the inverted information matrix of the r-subdesign is no greater than that of any other r-subdesign.
Several papers have been published on procedures for sequentially designing experiments using the criterion of D-optimality. See,
i
for example, Goldsmith (1974) and Wynn (1970). A common problem, however, was that the sequence had to start with a basic subdesign: one whose information matrix is non-singular. Published procedures were not helpful in choosing the best basic subdesign. It was in tackling this problem that I developed the algebra of section 2 of this chapter and hence to an algorithm based on the criterion of A-optimality. Unfortunately, this led to a computing problem: the computing time was too long for the algorithm to be of practical use. I therefore abandoned it and
returned to the D-optimality criterion. However, I am including 6 report of the algebraic development for the record, in case it may be useful elsewhere, and an outline of suggested algorithms.
Box and Draper (1971) pose practical arguments in favour of D-optimalityj 1. It forces experimenters to give some thought to the model to be postulated before experiments are actually done.
2. The number of observations is not restricted in any way so long as it is sufficient (the information matrix is non-singular). Extra observations can be added to the design so long as the experimenter chooses*
3. Since the search for the best design can be made over any specified region in the design variables, regions of special shape can be handled.
These arguments are particularly apposite with respect to asymmetric factorials. The first argument has been considered in earlier chapters when dealing with the design requirements (see algorithm ENFAC). The second answers the problem with which we opened this chapter: so long as we can find the best smallest basic subdesign, then we can add any number of points we like to it. The third argument also suits us.
The special shape of the region specified by our design variables, is that it must have as many dimensions as there are contrasts to be estimated and the variable represented in each of these dimensions can be set at one of only two values. This matter of coding the contrasts, and an algorithm to effect it, will be described fully in section three of this chapter. In that section I shall also present a new contribution to this field: an algorithm for choosing the best smallest basic
subdesign. This will be followed by the full sequential algorithm for building on to the basic subdesign and, in section four, some examples. Fortran listings are in appendix three.
2 Using the trace
The sum of the diagonal elements of a square matrix is called the trace of that matrix. A practical argument for the. A-optimality criterion is: When a set of linear coefficients is estimated by least squares, the variance of each estimated coefficient is proportional to the corresponding diagonal element of the inverted information matrix. Hence if the experiment is designed by choosing r observations such that the trace of the inverted information matrix is no greater than the corresponding trace for any other subset of r observations, we may fairly assume that the variance of each estimated coefficient is reasonably close to its minimum. Thus we may take the trace as a measure of the precision with which the coefficients may be estimated from the experimental results.
This leads to the simply stated algorithm: given that a basic subdesign has already been chosen, search among all observation points not yet included in the design for the point whose inclusion would cause the maximum reduction in the trace.
This still leaves the problem of choosing the best smallest basic subdesign, since the above algorithm relies upon the non-singularity of the information matrix so that it can have an inverse. I had an idea, however, that a generalised inverse might be used so that the search could begin as soon as a single row had been chosen as an anchor point for the design. If this were possible, as it proved to be, then the algorithm would become:
1. choose any point on the periphery of the design region (for example, if all the variables are coded (0,1) then choose the point 0 )} 2. although the information matrix of one design point is singular
so that the trace of its inverse is non-existent, assume that the trace does exist and let it be T;
3. using the concept of a generalised inverse, test every non-inoluded point in turn and find the point whose inclusion would cause the maximum reduction (AT), in T (it turns out that AT is a real • “ quantity);
4. repeat step 3 until a basic design is achieved;
3. use a modification of step 3 with a normal inverse,instead of a generalised inverse, to augment the basic design until it has the required number of points.
In developing the algebra to support this algorithm, I also considered the possibility of removing points from the design.
There are now four possibilities: 1. Stepping into a non-basic design; 2. Stepping out of a non-basic design; 3. Stepping into a basic design;
4. Stepping out of the basic design.
A flather requirement is clearly a test for the basicity of a design. This emerges from the theory developed to provide the four stepping procedures. The following initial relationships are needed:
1. Partitioned inverse
It is well established and easy to demonstrate that if a square matrix A is partitioned as A,„ ~11 ~12A., o A«. _~21 CMCM 7.1 if A has an inverse A ' 1,A S 7 A' 1 = -1 -1 -1 A ' + A, ' A,0 M ~11 ~11 ~12 ~ ~21 ~11 -in £12 i-1 -1 T /V -m"1a0, a7] L /S/ ~21 ->'11 M-1 7.2
where M = A,,„ - A„. A. . A.„~ ~22 ~>21 ~11 /v/12 2. Generalised inverse
If a non-basic, or singular, matrix A can be factored symmetrically
as A = uu' /V 7.3
where A is n-square and u is n by m, then a generalised inverse A* is defined as
— 1 —*1‘ 111
A* £ u (u'u) (u'u) u 1^ ^ A/ * ' A/ /V
so that A*A = u (u'u) ^ u 1AJ A/ AT AJ /V
7.4 7.5
It will be noted that although the rank of A is less than n,
since A is singular, provided the rank of A is greater than or equal to m then the rank of'(u'u) will be full so its inverse will exist. If, in fact, A is basic, or non-singular, it can be shown that the above definition (equation 7*4) satisfies the usual algebra of inversion. Suppose in this case that A can be similarly factored symmetrically:
A = uu1 7*3
so that A-1 = (uu-)"1rv ' /V(V 7.6
Let HI
S»
II '<1 7*7or 1 —i. II
7.8
then u'u(u’u) u' = u*A/ A/'/V /V' As 7*9
or A/ u 1 = u 1/sy 7.10
Thus the left hand side of 7*7 conforms with the right hand side
so that the relationships 7*6 and 7*7 apply equally well to all
factorable square matrices whether basic or non-basic► 3. Trace of a product
It is well known that if a square matrix A is a product of two